数学分析复习资料及公式大全.docx

导数公式:

= scc 2 x

/ 2 (cfgx)'= -cscr

(secx)r

= secx ・tgx (esc x\ = - esc x •

etgx (a x \ = a x \na

(arccosx)'=——/

yjl-x

2

— 2

I n = Jsin" xdx = jcos M

xdx 0

(log. x\ =

1

x\na

(arcctgx)f

=

1 l + x 2

基本积分表：

ygxdx = - ln|cos x +C ^ctgxdx = ln|sin x +C jscc xdx = ln|scc 兀 + fgx + C

Jese xdx = ln|csc x - etgx +

C 1 x =—arctg — +C a a = ±lnl dx

cos 2 x

dx

sin 2

x

|sec 2 xdx = tgx + C jese 2 xdx = -etgx + C

dx ~2 2

a +x dx 2 7 x -er

dx a 2

-x 2

dx

\la 2 -x 2

x-a

2ci \x + a\ 1 , ci + x 厂 =——In ---- + C

2a a-x

= = arcsin —+ C

a

jsec x • tgxdx = sec x + C |cscx-c/gxJx = -esex + C

ia x

dx = ———C

J

Inez

jshxdx = chx + C ^chxdx = shx + C

J 岛 T 777

"

^x 2

+a 2

dx = — y/x 2

+ a 2

+ — ln(x + y/x 2

+a 2

) + C 2

_________ ____________________ 2

JVx 2

-a 2d x =

~ J 兀2

_ — In 兀 + — cz 厶

+ C

JJ/

x = *罷 三角函数的有理式积分：

2 一 + — arcsin — + C

2

a sinx =

2u l +

u 2

cosx = 1 -M 2

1 + w 2

U=tg

2

dx =

2du

l + w 2

(arctgx)f = 1

l + x 2

/r 2

(arcsin x)f

sin(cr ± /?) = sin a cos /? ± cos <7

sin 0

cos(a ± 0) = cos a cos 0 年sin a sin

tga土tg0

•和差化积公式：

sirm + sin 0 = 2 sin cos ~~~~ sin 6Z-sin 0 = 2 cos °十

X _ -X

双曲正弦:shx=' r

2

双曲余弦:chx = C A

2

c/7r e x

双曲正切:thx = - = ^-^ chx e +e arshx = ln(x + Vx2 +1) archx = ± ln(x + y]x2 -1)

sinx

lim ----- =1

lim(l + -)' = ^ = 2.718281828459045... —8 %

arthx = —

In

2

三角函数公式:

•诱导公式：

数

角彳、

sin cos tg Ctg

-a-sina cosa-tga-ctga

90°-a cosa sina ctga tga

90°+a cosa-sina-ctga-tga

180°-a sina-cosa-tga-ctga

180°+a-sina・ cosa tga ctga

270°-a-cosa-sina ctga tga

270°+a-cosa sina-ctga-tga

360°-a-sina cosa-tga-ctga

360°+a sina cosa tga ctga •和差角公式:

tg(a±/3) =

/ , °、ctga・ctg0 +

\

crg(d±0)= & "

"sin ―—

2 2

r 6Z + 0 oc — (3 cos a + cos 0 = 2 n

cos —-^― cos —-^― cos a - cos 0

= 2 sin " + " sin —―—

2 2

•倍角公式:

•半角公式:

(济)(“)=£算严幼严

Jl=0

冲叫+和+

汕知” +…+⑷-1)・・°"+叽心)严+・・・ + "/)

2! k\

中值定理与导数应用：

拉格朗日中值定理:f(b)-f(a) = f^)(b-a) 柯西中值定理严)- W 以O

F(b)-F(a) F©

当F(Q 二x 时，柯西中值定理就是拉格朗口中值定理。 曲率：

sin 2a = 2 sin a cos a

cos2cr = 2cos 2 a-\-1 -2sin 2 a - cos 2 «-sin 2

a c c tg

2

a 一 1 ctg2a = ---------

2ctga

sin 3cr = 3 sin a -4sin 3

a cos3cr = 4COS '&-3COSQ

tg3a =

3tga tg'a \-3tg 2

a

• a

sin —=

2

a , /l-cos<7 1-COS6Z sin a tg — = ±A ------- = --------- = -------- 2 V 1 + coscr sin a 1 + COSQ

a , |l + cosQ

cos — = ±J ---------

2 V 2

a , Jl + cosa 1 + COSQ sin a ctg — = 土 J ----------------- = ----------- = ----------- 2 v 1-COS6Z sincr 1 -coscr

・正弦定理：-^— = -^— = ^— = 2R sin

A sin

B sinC

•余弦定理：c 2 = a 2 +b 2

-2abcosC

•反三角函数性质：arcsinx = ----- arccosx

2

71

arctgx = --- arcctgx

高阶导

莱布尼兹(Leibniz)公式: